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Irregularity of the Bergman projection on smooth unbounded worm domains (2204.05111v2)
Published 11 Apr 2022 in math.CV
Abstract: In this work we consider smooth unbounded worm domains $\mathcal Z_\lambda$ in $\mathbb C2$ and show that the Bergman projection, densely defined on the Sobolev spaces $H{s,p}(\mathcal Z_\lambda)$, $p\in(1,\infty)$, $s\ge0$, does not extend to a bounded operator $P_\lambda:H{s,p}(\mathcal Z_\lambda)\to H{s,p}(\mathcal Z_\lambda)$ when $s>0$ or $p\neq2$. The same irregularity was known in the case of the non-smooth unbounded worm. This improved result shows that the irregularity of the projection is not a consequence of the irregularity of the boundary but instead of the infinite windings of the worm domain.